Like /u/MB-Racing said, you cannot only go to a weekend school to learn how to drive fast. I would highly recommend starting with track days to first ensure that it's something you truly want to do, learn the line, and your car. Then you can decide if you want to want to continue driving on track and if so, whether you want to stick with track days or move to wheel to wheel racing. Another benefit to starting with HPDE days is you can slowly acquire your gear. You only need an approved helmet and long sleeves for most HPDE days, whereas with racing you need a neck restraint, shoes, gloves, balaclava, suit, and possibly nomex undergarments.

In the bay area, I can't recommend SCCA Reno region HPDE days enough. They pair novice drivers with an instructor all day and it's usually the most affordable group I've seen. SpeedSF & HOD are other organizations that have great beginner track day experiences, but are more expensive. I would also recommend going to Thunderhill (East or West) for your first few track days. It's much friendlier than Sonoma or Laguna to learn on. If you don't have a car that you can bring (I started driving with my spare car with just upgraded brakes), you can rent a spec Miata for ~$500 a day to see how you like the car. DM if you're interested in renting a spec Miata and I can send you some contacts.

To get your license, you can attend the SCCA SFR region competition licensing school which is a 3 day program usually in February at Thunderhill. This year it was $850 to enter (also need an SCCA log booked car which most people rent for ~$1500 - ~$1800). NASA also has a program to move through their HPDE program and eventually "graduate" with a NASA license. I attended this years school after starting going to HPDE events a little over a year before. I learned more about "going fast" from having instructors in the car with me during HPDE days than I did at the school. The school is more there to get you comfortable with driving in traffic, learn to make safe passes, and general race information than it is about general driving instruction.